ISIJ International, Vol. 61 (2021), No. 1

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* Corresponding author: E-mail: mxg_fy@163.com

© 2021 The Iron and Steel Institute of Japan. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

ISIJ International, Vol. 61 (2021), No. 1, pp. 55–61

https://doi.org/10.2355/isijinternational.ISIJINT-2019-799

1. Introduction

The security and stability operation of stave is the basis for a long-life operation of a BF. A good cooling system and process conditions, especially in the high heat load area of the waist and the bosh, can form a stable slag on the surface of stave, which can protect the stave and prolong the service life of BF. In recent years, domestic and foreign scholars have studied the heat transfer characteristics and slagging capacity of the copper stave thermal structure in the bosh region from the perspective of heat transfer.1–6) Y. Zhong et al.7) conducted a systematic feasibility analysis on the buried pure copper tubular cast copper stave. G. S. Guo et al.8) studied the heat transfer effect of cooling spe-cific surface area on the stave based on the finite element method. S. Lin et al.9) conducted a thermal test on a buried copper tubular stave, and concluded that the gas tempera-ture has a significant effect on the stress distribution of the copper stave body. Y. Deng et al.10) discussed the fracture mechanism of the rolled copper stave pipe and proposed a repairing method, K. Kawaoka et al.11) conducted a high

Optimization Analysis of Mechanical Performance of Copper Stave with Special-shaped Tubes in the Blast Furnace Bosh

Xiaogang MA1)* and Congcong WEN2)

1) School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan, Liaoning, 114051 China.2) Fuchunjiang Hydropwer Equipment Co, Ltd, Hangzhou, Zhejiang, 311100 China.

(Received on December 16, 2019; accepted on July 17, 2020; J-STAGE Advance published date: September 11, 2020)

Based on the theory of heat transfer, parametric modeling is established for the heat transfer model of copper cooling stave, which appears in recent years, with special-shaped tubes (elliptical, rectangular, double circular, three circular and ortho hexagonal) in a blast furnace (BF) bosh and the optimal tube for the cooling pipe is selected on the basis of the heat transfer characteristics of the stave. The heat transfer model of the hot end of stave embedded bricks which are not covered by slag, is analyzed using thermal-structural coupling method at the initial stage of blow-in under the normal working condition. The mutual influence of various parameters on the mechanical properties of copper stave is obtained using the response surface method. This method is combined with NSGA-II genetic algorithm to optimize the struc-ture parameters and longevity technology of the bosh. The optimized structure of the furnace bosh is improved in heat transfer characteristics and mechanical properties, which proves the model and param-eterized calculation program can be used as an optimized design and evaluation of the longevity technol-ogy of the bosh structure.

KEY WORDS: blast furnace bosh; special-shaped tubes; copper cooling staves; surface-response method; genetic algorithm.

temperature wear test on copper cooling stave copper mate-rial, and obtained that the strength of the copper cooling stave material would be significantly reduced in the envi-ronment exceeding 200°C. Recently, buried copper tubular copper stave has been successfully applied in many large BFs at home and abroad.12) However, water cracking and hot surface damage have occurred in copper stave, and the mechanism of this phenomenon lacks quantitative analysis. In addition, the previous research on copper stave is lim-ited to circular cooling water pipes or a single factor while lacks research on the mutual influence of various factors. Therefore, in this paper, based on the heat transfer theory, the heat transfer characteristics and mechanical properties of the copper stave that are commonly used in the BF bosh are analyzed. On the basis of optimizing the cooling water pipe type, the thickness of the copper stave body, the cool-ing specific surface area, the cooling water speed and the cooling water temperature are compared. The maximum thermal stress of the stave body under the normal working condition of the BF is used as an index to quantitatively calculate the correlation between each parameter and the index parameters. Taking the maximum thermal stress of the copper stave body as the optimization target, the structural

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parameters and longevity technology of the thermal struc-ture of the bosh area are optimized, and the heat transfer model of the bosh structure before and after optimization is compared and analyzed using a combination of response surface model and NSGA-II genetic algorithm method, which verifies the effectiveness of this method to optimize the thermal structure of BF.

2. Boundary Conditions and Finite Element Model

The analysis is carried out for the structure of the bosh area of a BF,13) which consists of the furnace shell (includ-ing bolts, locating pins, etc.), the packing layer, copper stave (including the cooling water pipes, the stave of the BF bosh is a circular water pipe in this paper) and embedded bricks, from the outside to the inside. The gap between adjacent stave walls and between the stave and the shell is filled with filler. The stave is a “four in and four out” buried pure copper tube cast copper stave. Considering the fact that the working environment of each stave is basically the same and for the purpose of reducing the calculation amount, a single stave + side gap filler + packing layer + furnace shell are taken as the calculating model. The upper and lower end faces of the stave have the same displacement and the circumferential displacement of the left and right sides is zero. The monolithic stave will have periodic mechanical characteristics when the heat transfer between the upper and lower sides is neglected. The temperature gradient of the upstream and downstream of the stave caused by the

water temperature difference under working conditions has little influence on the coupling characteristics thus the aver-age temperature is taken to reduce the calculation amount. The sector structure model of the single block stave in the cylindrical coordinate system is shown in Fig. 1.

The expression of the integrated convective heat transfer coefficient hs between the furnace shell and the surrounding air14) is:

h Ts � � �9 3 0 058 1. . ........................... (1)

Where T1 is the ambient temperature around the furnace shell, °C

According to the research data of L. J. Wu et al.15) based on the boundary condition substitution method, which determined that the comprehensive convective heat transfer coefficient hx between the hot furnace gas and the tile hot surface in the range of 500–1 248°C varies with temperature as shown in Eq. (2).

h T Tx � � � � � �5 606 0 207 8 414 1025

22. . . .......... (2)

Where T2 represents the hot furnace gas temperature, °C.The copper stave has the advantages of high thermal con-

ductivity, stable dross, high load and long life. The buried copper tube cast copper stave avoids the welding process and completely eliminates the air gap layer. The structural size parameters of each layer in the bosh area of a blast furnace are shown in Table 1.

3. Heat Transfer Characteristics Analysis of Various Tubular Copper Stave

In order to better compare the influence of each tube type on the heat transfer characteristics of the stave, the circular tube is taken as the reference standard, and the heat transfer characteristics of the stud tube stave in the two cases are considered, that is, the same cross-sectional area and the same circumference as the circular tube.

3.1. Heat Transfer Characteristics Analysis of the Stave with Same Cross-sectional Area

The equivalent diameter DG of each shaped tube is

D A SG = 4 / ................................ (3)

Where A represents the cross-sectional area; S represents the wet perimeter length.

The equivalent diameter of each shaped tube is shown in Table 2. The ratio of the short axis of the rectangular tube

Table 1. Dimension parameters of stave.

Parameter name Numerical value/mm Parameter name Numerical value/mm

Thickness of furnace shell 40 Distance between pipe and hot end (a) 140

Packing layer thickness 50 Distance of pipe end face (b) 80

Thickness of cooling stave ( f ) 180 Distance between pipes (e) 179.25

Height of cooling stave 1 385 Side clearance width 20

Width of cooling stave 717 Depth of dovetail slot (c) 60

Width of brick (d) 100 Width of dovetail slot 80×100

Pipe diameter 45 Cooling plate inner radius 4 000

Fig. 1. Three-dimensional physical model of the BF bosh. (Online version in color.)

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to the elliptical tube b/a is calculated as 0.6.The relationship between the inertial force and the vis-

cous force of a fluid in a given flow field can be character-ized by the Reynolds number Re, the expression is

Re l l� ��� � � �/ / ......................... (4)

Where v is the fluid velocity; υ is the fluid kinematic viscos-ity; ρ is the fluid density; μ is the hydrodynamic viscosity; and l is the characteristic length.

The Reynolds number is the basis for judging the fluid flow state. For a circular tube, the fluid flow state will be laminar flow when Re≤2 300; it will be turbulent flow when Re≥104; and it will be transition flow when 2 300< Re<104. Taking a circular tube as an example, the convective heat transfer coefficient of the turbulent cooling water can be expressed by the Dietes and Bolt characteristic correlation16) for a smooth or vertical long tube with a constant wall temperature and a small temperature difference between the fluid and the wall.

Nu Re Prn= 0 023 0 8. . .......................... (5)

Where n is the index, n = 0.4 when the fluid is heated; Pr is the Prandtl number.

Combining Eqs. (4), (5) and Nusselt number Nu=h·DG/k, the convective heat transfer coefficient expression of cool-ing water in a circular tube under turbulent flow conditions is obtained:

h v= 3 899 750 0 8. . ............................ (6)

According to this, the expressions of the convective heat transfer coefficient h of each type of internal cooling water are obtained, as shown in Table 3.

When the cooling water flows in each tube type, the pump power and the water flow velocity are the same. The enhanced heat transfer evaluation criterion can be used as the evaluation standard for each type of enhanced heat transfer. The calculation formula is

� � � � � �Nu Nu f fi i/ / //

0 01 3 .................... (7)

Where Nui and Nu0 are the Nusselt number of the shaped tube and the standard tube respectively; fi and f0 are the resistance coefficients of the shaped tube and the standard tube respectively. The coefficient of resistance of the tube can be obtained by the Blasius formula.17)

The hot furnace gas temperature is 1 200°C. Figure 2 is a graph showing the maximum temperature Tmax of each tubular copper stave body in the bosh region of the BF under the same cross-sectional area and cooling water flow rate of 3 m/s.

It can be seen from the calculation results in the figure that the Tmax of each shaped tube has a slight decrease compared with that of the circular tube with the same

cross-sectional area, and the temperature of cooling stave with rectangular tubes (b/a=0.6) and cooling stave with three circular tubes have the most obvious decrease. Fig-ure 3 shows the variation of Tmax and η in different b/a of rectangular tubes. As b/a increases, Tmax increases continu-ously, and η decreases continuously. When b/a is greater than 0.6, Tmax increases sharply while η drop tends to be flat. Once the copper cooling stave is damaged, the only way to replace it with a new one is to shut down the fur-

Table 2. Equivalent diameter of section tube.

Name Circular tube Rectangular tube Elliptical tube

DG/mm 45.000 38.614 42.270

Name Double circular tube

Three circular tube

Ortho hexagonal tube

DG/mm 33.750 30.000 42.854

Table 3. Expression of convective heat transfer coefficient.

Name Circular tube Rectangular tube Elliptical tube

H h=3 899.75v0.8 h= 4 020.97v0.8 h=3 948.87v0.8

Name Double circular tube

Three circular tube

Ortho hexagonal tube

H h= 4 127.81v0.8 h= 4 154.17v0.8 h=3 938.05v0.8

Fig. 2. Tmax (v =3 m/s) for each tube type under the same cross-sectional area.

Fig. 3. Tmax and η distribution of Rectangular tubes with different b/a.

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Fig. 7. The influence results of various parameters on Smax.

Fig. 8. The optimized stress field of copper stave.

Fig. 4. Temperature field of each structure of the Copper Stave with rectangular tubes (b/a= 0.6).

nace, which can cause a significant productivity reducing of the enterprise. The main reason for the damage of the copper cooling stave is the high temperature and repeated deformation (mainly caused by the repeated change of temperature difference). Therefore, the pursuit of lower

temperature is the primary consideration for the selection of cooling water tube parameters. From the calculation of the enhanced heat transfer evaluation criteria and the flow resistance characteristics of fluid, it can be seen that the flatter the cooling water tube is, the greater the resistance of the cooling water tube to the cooling water will be and thus under the same cooling water flow speed, the pressure at the inlet end of the water tube needs to be increased to improve the power of the water pump. At the same time, the flatter the cooling water tube means that the smaller the radial bearing capacity of the copper cooling wall is, and the more difficult the processing is. Therefore, in pur-suit of a lower temperature, the evaluation criterion η of enhanced heat transfer can be appropriately reduced, which not only saves the cost of the enterprise, but also increases the deformation resistance of the cooling stave. Accord-ing to the analysis results of heat transfer characteristics of copper cooling stave, the intersection of the maximum temperature Tmax of copper cooling stave with rectangular tube and the evaluation criterion η of enhanced heat trans-fer should be the optimal tube type size. At this time, the rectangular tube b/a=0.62, as shown in Fig. 3. Figure 4 is a temperature field cloud diagram of the thermal structure of the rectangular section of the rectangular tube (b/a = 0.6).

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3.2. Heat Transfer Characteristics Analysis of Stave on the Same Perimeter

Figure 5 shows the calculation results of Tmax for each tube type with the same circumference and cooling water speed of 3 m/s. It can be seen from it that the temperature of the stave with elliptical tubes decreases the most with the same circumference. Figure 6 shows the distribution of Tmax and η under different b/a of elliptical tubes. From the perspective of Comprehensive consideration from the com-prehensive evaluation index of heat transfer enhancement, the temperature of cooling stave, enterprise cost, etc., the optimal tube type for each tube type with the same circum-ference should be an elliptical tube with b/a=0.57.

Analyzing the calculation results of Figs. 2 and 5, it can be obtained that the copper stave of the rectangular tube has a small difference in heat transfer characteristics compared with the copper stave with elliptical tubes in the same cross-sectional area while has a slightly lower value in terms of enhanced heat transfer evaluation indicators. Considering the heat transfer, flow resistance characteristics and the stress concentration point of the rectangular tube during working process, the cooling water pipe of copper stave in the BF bosh area will be most suitable to change to an elliptical tube with b/a=0.57. At the same time, compared with the original stave with circular pipes, it can save a large amount of cooling water and improve efficiency while achieving the same cooling effect.

4. Optimization of Structure Longevity Technology in Blast Furnace Bosh Area

4.1. Introduction to Response Surface MethodThe basic idea of response surface methodology is to

express implicit functional functions by constructing poly-nomials with well-defined forms. Assume that the relation-ship between the performance function Z and the random variable Q= [Q1, Q2, ..., QR] is as shown in Eq. (8). The N samples of the random variable are randomly sampled, and the performance function values (z1, z2, ..., zN) are statisti-cally analyzed, and the system function is obtained by least squares theory fitting.18)

Z a a Q a QQi i

i

R

ij i j

j i

R

i

R

� � �� ��� ��0

1 1

................ (8)

Where a0, ai, aij (i=1,..., R; j= i, ..., R) are the undeter-mined coefficients of the functional equation, for a total of 1+R+R(R+1)/2.

The matrix method is used to take three horizontal points for each random variable and then according to the Box-Behnken (BBD) sampling method, the center and the middle midpoint are taken as the sample value points. When the random variable distribution conforms to a certain law, the variable level qs can be determined by Eq. (9).

f q dq p nn

qs( ) , , ,� �

��� 1 2 3 ..................... (9)

Where f(q) represents a random variable probability density function. pn represents a horizontal point, taking p1=0.01, p2=0.5, and p3=0.99.

Proceeding numerical simulation of the sample values

of S random variables yields S output points (z1, z2, ..., zS). Then the Eq. (10) can be obtained using regression analysis, and the functional relationship of the performance function (8) can be determined. The design method used is the BBD method.

S z a a q a q qi i i ij i j

j i

R

i

R

i

R

i

S

� � � ��

���

�

���

�

���

�

�������

���� 0

111

2

..... (10)

4.2. Maximum Thermal Stress Response Surface Model of Copper Stave Body

Thermal stress is one of the key factors affecting the service life of the BF stave. From the numerical calcula-tion of the thermal stress of the stave body, the influence law is analyzed when each parameter varies, which has far-reaching significance for the longevity of the BF. Based on the structure of blast furnace bosh in Table 1, the maxi-mum thermal stress Smax (ie Von Mises equivalent stress)

Fig. 5. Tmax (v =3 m/s) for each tube type under the same circum-ference.

Fig. 6. Tmax and η distribution of elliptical tubes with different b/a.

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of the copper cooling stave with elliptical tube which has the same circumference with circular tube and a short-axis ratio of 0.6 is taken as research object, and the thickness of the copper stave body, the cooling specific surface area, the cooling water flow rate and the cooling water temperature as comparative sequences. The parameters are shown in Table 4.

Multivariate regression fitting analysis is performed on the Smax calculation results, and Table 5 is the results. The model determines the coefficient R2=0.9463 and the adjust-ment decision coefficient R2adj=0.8925, which indicates that the calculation result of the model is well fitted to the exper-imental data, and can be used for the theoretical prediction of the maximum thermal stress Smax of the copper stave body with various parameters. The significance of each parameter to Smax is determined by the F test: the smaller the P value and the larger the F value, which means the higher the degree of significance. It can be seen from Table 5 that the influence of x2 and x4 on Smax is particularly prominent, and the interaction of x1 and x2 in the interaction term has a greater influence on Smax. Due to space limitations, results of the influence rule of partial interaction parameters on Smax are shown in Fig. 7.

The degree of influence of each parameter on Smax ranges from largest to smallest: x2, x4, x3, x1, and the interaction between x1 and x2 is relatively large. According to the response surface theory, the greater the degree of influ-ence is, the greater the influence of this factor on the target sequence will be. Smax increases firstly and then decreases with the increase of x1 and based on this, when the value of x2is small, the thickness of the stave should be moder-ately reduced. Smax decreases rapidly with the increase of x2 and then becomes gentle. When the specific surface area increases from 0.7 to 1.1, Smax drops sharply, and based on this, the specific surface area can be appropriately increased to reduce the cooling wall stress. When x3 is increased from 2 m/s to 4 m/s, the Smax decreases slightly because the heat transfer resistance of heat convection of cooling water is not the limiting element of the heat transfer of the stave any more. Therefore, an improvement in the cooling water velocity means an increase in the water flow rate, and at the same time it will lead to a significant increase in the water flow resistance and the running cost. When the value of x4 is relatively small, the water speed can be reduced to reduce the cost but increased to reduce the cooling wall stress when the water temperature is high. In the actual operation, atten-tion should also be paid to the repeated thermal stress of

the body caused by repeated changes in water temperature which is easy to cause fatigue damage, so the water tem-perature pursuit is mainly based on stability.

According to formula (10), the response surface equation of the maximum thermal stress Smax of the copper stave body during normal operation of the blast furnace is

S x x x

x

max � � � �� �

�421 8408 6 44753 280 47565

0 52697 1

38 10621 2 3

4

. . .

.

.

.. . .

. .

95817 0 02244 1 729

10 5 85167 8 83333

1 2 1 3

31 4 2 3

x x x x

x x x x

� �

� � �� ��

� � �

�

�

�

�

10

10 0 013591 2 33426

3 37783

4 525

34

34 1

22

2

3

2

3

x x

x x x x

x

. . .

. 224

20 01268� . x

........................................ (11)

4.3. Target Optimization Based on Genetic AlgorithmBased on the response surface model of the copper stave

bosh of the furnace, the genetic algorithm is used to opti-mize the maximum thermal stress of the body under multi-parameters condition.19) Take the optimized overall mass mT of the copper stave no higher than 95% of the initial mass and the water consumption Ls per unit time no higher than 95% of the water consumption in the initial unit time as the constraint condition.

The mathematical model of the optimized design can be expressed as

S S x x x x

x x

x

y �

� �

min( ( , , , ))

( , ), ( . , . ),

max 1 2 3 4

1 2

3

155 205 0 7 1 3s.t.

�� �� �

( , ), ( , )

. , .

2 4 20 40

0 95 0 954

0 0

x

m m L LsT

.............. (12)

Where ρ represents the copper stave material density; m0 represents the initial mass of the elliptical tube copper stave

Table 5. Calculations of response surface methodology.

Random variables

Quadratic sum

Degree of freedom

Mean square F P (P >F)

x1 432.55 1 432.55 5.25 0.0379

x2 2 646.89 1 2 646.89 32.15 0.0001

x3 732.44 1 732.44 8.90 0.0099

x4 1 022.39 1 1 022.39 12.42 0.0034

x1x2 862.74 1 862.74 10.48 0.0060

x1x3 1.26 1 1.26 0.02 0.9033

x1x4 0.75 1 0.75 9.08×10 −3 0.9254

x2x3 12.33 1 12.33 0.15 0.7046

x2x4 2.81×10 −3 1 2.81×10 −3 3.41×10 − 5 0.9954

x3x4 8.19×10 −3 1 8.19×10 −3 9.95×10 − 5 0.9922

x12 440.18 1 440.18 5.35 0.0365

x22 0.29 1 0.29 3.48×10 −3 0.9538

x32 74.01 1 74.01 0.90 0.3592

x42 8.45 1 8.45 0.10 0.7534

Residual error 1 152.64 14 82.33

Sum 7 497.06 28

Table 4. Levels of each parameter.

Random parametersHorizontal encoding value

−1 0 1

x1/mm 155 180 205

x2 0.7 1.0 1.3

x3/m·s −1 2 3 4

x4/°C 20 30 40

Note: x1, x2, x3 and x4 are the copper stave body thickness, cooling specific surface area, cooling water flow rate and cooling water temperature, respectively.

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Table 6. Calculation results before and after optimization.

Name x1/mm x2 x3/m·s −1 x4/°C Smax/MPa mT /kg Ls/m3

Initial value 180 1.0 3.0 30 205.2 0.172ρ 0.012

Optimal value 171 1.1 2.3 30 197.2 0.162ρ 0.011

and L0 represents the initial unit time water consumption.The genetic algorithm has a crossover probability of 0.6

and a mutation probability of 0.01. The calculation results of each random variable and objective function before and after optimization are shown in Table 6.

In order to verify the accuracy of the optimized results, the optimized parameter values are brought into the finite element model, and the results are shown in Fig. 8. The comparison results of the response surface equation are 199.09818 MPa, and the approximation error is 0.9549%, which indicates that the response surface model and the NSGA-II genetic algorithm have higher precision for opti-mizing the structural parameters and longevity technology of the BF bosh area.

5. Conclusion

(1) From the perspective of the temperature of cooling stave, the comprehensive evaluation index of heat transfer enhancement, enterprise cost, machining difficulty, radial bearing capacity of cooling stave, etc., the optimal tube type for each tube type should be an elliptical tube with b/a=0.57. After the cooling water pipe of the copper stave in the bosh area is changed to the elliptical tube, the cooling effect can be ensured while the cooling water consumption is saved, which can help to effectively improve the enter-prise efficiency.

(2) The degree of influence of each parameter on Smax is as follows: cooling specific surface area, cooling water temperature, cooling water velocity and thickness of the stave body. Interaction between the thickness of the stave

body and the cooling specific surface area is relatively large in terms of interaction. When designing the structure of the bosh area, the influence of various factors on the distribution of the stress field of the stave can be referenced to ensure the safe operation of the stave while optimizing its structure.

(3) The objective function is optimized by the combi-nation of response surface method and NSGA-II genetic algorithm. The optimized stave is improved in heat transfer characteristics, and the mass of the body and the amount of cooling water are reduced. The effectiveness of the optimi-zation method of the thermal structure parameters of the BF bosh area combined with the response surface method and the genetic algorithm is verified.

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